# Can I state the Mandelbrot Set/Equation like this?

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1. May 6, 2015

### emergentecon

f(z) = z2 + c

2. May 6, 2015

### Raghav Gupta

But it must be under iteration. Then definitely you can do that.
Here I suppose you are saying z and c are complex numbers.

3. May 6, 2015

### emergentecon

Correct yes . . . under iteration, and z + c are complex numbers.
Thanks!

4. May 6, 2015

### HallsofIvy

Staff Emeritus
Just the simple equation "$f(z)= z^2+ c$" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "$z_{n+1}= z_n^2+ c$, $z_0= 0$"

5. May 6, 2015

### emergentecon

I can't say this helps me much.

Is it wrong to say that if you iterate the function f(z) = z^2 + c over the complex numbers, that you get the mandelbrot set?

6. May 6, 2015

### emergentecon

For instance, if I wanted to implement the recursion in Excel, I would in essence specify it as f(z) = z^2 + c
The mandelbrot set is often 'summarised' as z = z^2 + c so why is it wrong to write f(z) = z^2 + c ????

7. May 6, 2015

### HallsofIvy

Staff Emeritus
??? Just "iterating a function" will give you a lot of numbers and a lot of points where the iteration does not converge to any number. Again, the "Mandlebrot set" is the set of complex numbers for which the iteration does converge.

I don't know what you mean by "summarized" here. What do you do with that iteration to get the Mandlebrot set. (The "Julia sets" use that same iteration but in a different way.)

8. May 6, 2015

### stedwards

Given f(z) = z2 + c then zn+1 = f(zn). z0=0, c ranges.

The mandelbrot set is over the domain, c.

If you want to produce a mandelbrot set, you need more than an equations. You need an algorithm.

Last edited: May 6, 2015
9. May 6, 2015

### micromass

Staff Emeritus
It all depends on context. If you are talking to somebody about dynamical systems, then yes, you can write it like that. If you're not in the correct context, then no, you can't. It all depends on whether the other person will understand you or not.

10. May 8, 2015

### emergentecon

I mis-stated my question, meant to say his equation, as opposed to the set.
As I know, when he was asked about his work, he wrote down the equation as: z -> z^2 + c
So was wondering if, in this context, z = z^2 + c or f(z) = z^2 + c as opposed to z(n+1) = z(n)^2 + c

11. May 8, 2015

### Staff: Mentor

I would bet that it was z <- z2 + c, with the idea being that, starting with a specified complex number c and some complex number z, you square z, add c, and use that as your new z. Then repeat. And repeat. Ad infinitum.
The latter formula better captures the idea of an endless sequence of complex numbers.

12. May 8, 2015

Thanks!